Go to the Illinois DNR website and check the non-resident archery tag date, it is usually in early june. On the day they go on sale you can purchase your tag directly online, there is no drawing. The tag presently cost $400 and you also need to purchase a hunting license and a habitat stamp. The total will run you around $450.
I have hunted Champaign and Vermillion counties for the last 21 years. I always hunt for 6 days and usually start around the 7th of November. I hope this helps.